We present B-GRASP (Bayesian GRAph inference with SPDE priors), a Bayesian framework for inferring uncertain edge weights in stochastic dynamical systems on graphs from noisy observations of nodal states. The unknown edge weights parameterize the graph differential operator and therefore directly govern the evolution of the graph process. Motivated by connections between differential operators in PDEs and SPDEs and their graph counterparts, we construct stochastic graph models incorporating diffusion, reaction dynamics, and stochastic forcing. The resulting hierarchical formulation jointly represents uncertainty in the graph structure and stochastic forcing. Latent graph variables determine positive edge weights and the corresponding graph Laplacian, while latent Brownian variables represent the stochastic forcing. Conditional on these variables, the graph dynamics define a deterministic forward map from which the likelihood and posterior distribution are constructed. We characterize the posterior using maximum a posteriori estimation and the No-U-Turn Sampler, enabling both point estimation and uncertainty quantification. We demonstrate the framework on a one-dimensional inverse heat-conduction problem, stationary and nonstationary graph reaction-diffusion systems with nonlinear dynamics, and state-level COVID-19 data in the United States. The numerical results show that posterior uncertainty provides information not captured by point estimates, particularly for weakly identifiable or highly conductive edges, and enables uncertainty in both graph connectivity and stochastic forcing to be quantified within a Bayesian framework.

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